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href="javascript:void(0);"><i class="fas fa-bars fa-fw"></i></a></div></div></nav><div id="post-info"><h1 class="post-title">Diffie-Hellman的C++语言描述简单实现</h1><div id="post-meta"><div class="meta-firstline"><span class="post-meta-date"><i class="far fa-calendar-alt fa-fw post-meta-icon"></i><span class="post-meta-label">发表于</span><time class="post-meta-date-created" datetime="2022-11-16T05:37:26.000Z" title="发表于 2022-11-16 13:37:26">2022-11-16</time><span class="post-meta-separator">|</span><i class="fas fa-history fa-fw post-meta-icon"></i><span class="post-meta-label">更新于</span><time class="post-meta-date-updated" datetime="2023-04-26T07:46:22.833Z" title="更新于 2023-04-26 15:46:22">2023-04-26</time></span><span class="post-meta-categories"><span class="post-meta-separator">|</span><i class="fas fa-inbox fa-fw post-meta-icon"></i><a class="post-meta-categories" href="/categories/%E7%BD%91%E7%BB%9C%E5%AE%89%E5%85%A8/">网络安全</a></span></div><div class="meta-secondline"><span class="post-meta-separator">|</span><span class="post-meta-pv-cv" id="" data-flag-title="Diffie-Hellman的C++语言描述简单实现"><i class="far fa-eye fa-fw post-meta-icon"></i><span class="post-meta-label">阅读量:</span><span id="busuanzi_value_page_pv"><i class="fa-solid fa-spinner fa-spin"></i></span></span></div></div></div></header><main class="layout" id="content-inner"><div id="post"><article class="post-content" id="article-container"><h1 id="前言"><a href="#前言" class="headerlink" title="前言"></a>前言</h1><p>网络安全中Diffie-Hellman的C++语言描述简单实现。</p>
<hr>
<h1 id="代码仓库"><a href="#代码仓库" class="headerlink" title="代码仓库"></a>代码仓库</h1><ul>
<li><a target="_blank" rel="noopener" href="https://github.com/yezhening/Programming-examples">yezhening&#x2F;Programming-examples: 编程实例 (github.com)</a></li>
<li><a target="_blank" rel="noopener" href="https://gitee.com/yezhening/Programming-examples">Programming-examples: 编程实例 (gitee.com)</a></li>
</ul>
<hr>
<h1 id="代码特点"><a href="#代码特点" class="headerlink" title="代码特点"></a>代码特点</h1><p>纯C++语言：</p>
<ul>
<li>相对规范和整洁</li>
<li>一定程度地面向对象</li>
<li>使用一部分高级特性</li>
<li>考虑优化性能</li>
</ul>
<p>详细注释：</p>
<ul>
<li>提示规范和整洁</li>
<li>提示面向对象</li>
<li>提示高级特性</li>
<li>提示优化性能</li>
<li>解析Diffie-Hellman步骤（<strong>网络上大部分实现代码的含义不明确，本代码相对明确</strong>）</li>
<li>注意易错点</li>
</ul>
<hr>
<h1 id="代码"><a href="#代码" class="headerlink" title="代码"></a>代码</h1><h2 id="dh-h"><a href="#dh-h" class="headerlink" title="dh.h"></a>dh.h</h2><figure class="highlight cpp"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">#<span class="keyword">ifndef</span> DH_DH_H_</span></span><br><span class="line"><span class="meta">#<span class="keyword">define</span> DH_DH_H_</span></span><br><span class="line"></span><br><span class="line"><span class="meta">#<span class="keyword">include</span> <span class="string">&lt;iostream&gt;</span> <span class="comment">//cout、endl</span></span></span><br><span class="line"></span><br><span class="line"><span class="keyword">using</span> std::cout;</span><br><span class="line"><span class="keyword">using</span> std::endl;</span><br><span class="line"></span><br><span class="line"><span class="comment">// Diffie-Hellman类</span></span><br><span class="line"><span class="keyword">class</span> <span class="title class_">DH</span></span><br><span class="line">&#123;</span><br><span class="line"><span class="keyword">public</span>:</span><br><span class="line">    <span class="built_in">DH</span>();                         <span class="comment">// 构造</span></span><br><span class="line">    <span class="function"><span class="type">unsigned</span> <span class="type">int</span> <span class="title">GetPrivateKey</span><span class="params">()</span></span>; <span class="comment">// 获取私钥</span></span><br><span class="line">    <span class="function"><span class="type">unsigned</span> <span class="type">int</span> <span class="title">GetPublicKey</span><span class="params">(<span class="type">const</span> <span class="type">unsigned</span> <span class="type">int</span> &amp;x)</span></span>; <span class="comment">// 获取公钥</span></span><br><span class="line">    <span class="function"><span class="type">unsigned</span> <span class="type">int</span> <span class="title">GetKey</span><span class="params">(<span class="type">const</span> <span class="type">unsigned</span> <span class="type">int</span> &amp;y, <span class="type">const</span> <span class="type">unsigned</span> <span class="type">int</span> &amp;x)</span></span>; <span class="comment">// 获取密钥</span></span><br><span class="line"></span><br><span class="line"><span class="keyword">private</span>:</span><br><span class="line">    <span class="comment">// 提示：算法未明确说是正数，但大多使用正数</span></span><br><span class="line">    <span class="function"><span class="type">unsigned</span> <span class="type">int</span> <span class="title">GetPrimeNum</span><span class="params">()</span></span>;                                                                    <span class="comment">// 获取素数</span></span><br><span class="line">    <span class="function"><span class="type">bool</span> <span class="title">PrimalityTest</span><span class="params">(<span class="type">const</span> <span class="type">unsigned</span> <span class="type">int</span> &amp;n, <span class="type">const</span> <span class="type">unsigned</span> <span class="type">int</span> &amp;a)</span></span>;                              <span class="comment">// Miller-Rabin素性测试</span></span><br><span class="line">    <span class="function"><span class="type">unsigned</span> <span class="type">int</span> <span class="title">QuickPowMod</span><span class="params">(<span class="type">const</span> <span class="type">unsigned</span> <span class="type">int</span> &amp;a, <span class="type">const</span> <span class="type">unsigned</span> <span class="type">int</span> &amp;q, <span class="type">const</span> <span class="type">unsigned</span> <span class="type">int</span> &amp;n)</span></span>; <span class="comment">// 蒙哥马利快速幂模运算</span></span><br><span class="line">    <span class="function"><span class="type">unsigned</span> <span class="type">int</span> <span class="title">QuickMulMod</span><span class="params">(<span class="type">const</span> <span class="type">unsigned</span> <span class="type">int</span> &amp;a, <span class="type">const</span> <span class="type">unsigned</span> <span class="type">int</span> &amp;b, <span class="type">const</span> <span class="type">unsigned</span> <span class="type">int</span> &amp;c)</span></span>; <span class="comment">// 快速乘</span></span><br><span class="line">    <span class="function"><span class="type">unsigned</span> <span class="type">int</span> <span class="title">GetPrimitiveRoot</span><span class="params">()</span></span>;                                                               <span class="comment">// 获取本原根</span></span><br><span class="line"></span><br><span class="line">    <span class="type">unsigned</span> <span class="type">int</span> p_arg_; <span class="comment">// 参数p</span></span><br><span class="line">    <span class="type">unsigned</span> <span class="type">int</span> a_arg_; <span class="comment">// 参数a</span></span><br><span class="line">&#125;;</span><br><span class="line"></span><br><span class="line"><span class="meta">#<span class="keyword">endif</span> <span class="comment">// DH_DH_H_</span></span></span><br></pre></td></tr></table></figure>
<hr>
<h2 id="dh-cpp"><a href="#dh-cpp" class="headerlink" title="dh.cpp"></a>dh.cpp</h2><figure class="highlight cpp"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span 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class="line"><span class="meta">#<span class="keyword">include</span> <span class="string">&lt;ctime&gt;</span>   <span class="comment">//time()</span></span></span><br><span class="line"><span class="meta">#<span class="keyword">include</span> <span class="string">&lt;cstdlib&gt;</span> <span class="comment">//srand()、rand()</span></span></span><br><span class="line"></span><br><span class="line"><span class="meta">#<span class="keyword">include</span> <span class="string">&quot;dh.h&quot;</span></span></span><br><span class="line"></span><br><span class="line"><span class="comment">// 构造</span></span><br><span class="line">DH::<span class="built_in">DH</span>()</span><br><span class="line">&#123;</span><br><span class="line">    <span class="comment">// 全局公开量：</span></span><br><span class="line">    <span class="comment">//  1. 选择p。p为素数</span></span><br><span class="line">    <span class="comment">//  注意：将随机种子提取放在循环外、相同函数外，以避免时间相近获取的随机数相同</span></span><br><span class="line">    <span class="type">unsigned</span> <span class="type">int</span> seed = <span class="built_in">time</span>(<span class="literal">nullptr</span>); <span class="comment">// 随机种子</span></span><br><span class="line">    <span class="built_in">srand</span>(seed);                       <span class="comment">// 设置随机种子</span></span><br><span class="line"></span><br><span class="line">    <span class="keyword">this</span>-&gt;p_arg_ = <span class="keyword">this</span>-&gt;<span class="built_in">GetPrimeNum</span>(); <span class="comment">// 获取素数</span></span><br><span class="line"></span><br><span class="line">    <span class="comment">// 2.计算a。a &lt; p且a是p的本原根</span></span><br><span class="line">    <span class="keyword">this</span>-&gt;a_arg_ = <span class="keyword">this</span>-&gt;<span class="built_in">GetPrimitiveRoot</span>(); <span class="comment">// 获取本原根</span></span><br><span class="line"></span><br><span class="line">    cout &lt;&lt; <span class="string">&quot;全局公开量：&quot;</span> &lt;&lt; endl;</span><br><span class="line">    cout &lt;&lt; <span class="string">&quot;参数p：\t&quot;</span> &lt;&lt; <span class="keyword">this</span>-&gt;p_arg_ &lt;&lt; endl;</span><br><span class="line">    cout &lt;&lt; <span class="string">&quot;参数a：\t&quot;</span> &lt;&lt; <span class="keyword">this</span>-&gt;a_arg_ &lt;&lt; endl;</span><br><span class="line">    cout &lt;&lt; endl;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="comment">// 获取私钥</span></span><br><span class="line"><span class="function"><span class="type">unsigned</span> <span class="type">int</span> <span class="title">DH::GetPrivateKey</span><span class="params">()</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="comment">// 私钥要求：随机整数；简化为非负数；小于参数p</span></span><br><span class="line">    <span class="type">unsigned</span> <span class="type">int</span> random = <span class="number">0</span>; <span class="comment">// 随机数</span></span><br><span class="line"></span><br><span class="line">    <span class="keyword">while</span> (<span class="number">1</span>) <span class="comment">// 循环</span></span><br><span class="line">    &#123;</span><br><span class="line">        random = <span class="built_in">rand</span>(); <span class="comment">// 随机数 一般是4~5位数，不超过unsigned int的表示范围</span></span><br><span class="line"></span><br><span class="line">        <span class="keyword">if</span> (random &lt; <span class="keyword">this</span>-&gt;p_arg_)</span><br><span class="line">        &#123;</span><br><span class="line">            <span class="keyword">break</span>;</span><br><span class="line">        &#125;</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="keyword">return</span> random;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="comment">// 获取公钥</span></span><br><span class="line"><span class="function"><span class="type">unsigned</span> <span class="type">int</span> <span class="title">DH::GetPublicKey</span><span class="params">(<span class="type">const</span> <span class="type">unsigned</span> <span class="type">int</span> &amp;x)</span> <span class="comment">// 参数：私钥</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="comment">// 公钥Y = 参数a的私钥X次方 mod p（X是以a为底的模p离散对数）</span></span><br><span class="line">    <span class="type">unsigned</span> <span class="type">int</span> y = <span class="keyword">this</span>-&gt;<span class="built_in">QuickPowMod</span>(<span class="keyword">this</span>-&gt;a_arg_, x, <span class="keyword">this</span>-&gt;p_arg_);</span><br><span class="line"></span><br><span class="line">    <span class="keyword">return</span> y;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="comment">// 获取密钥</span></span><br><span class="line"><span class="function"><span class="type">unsigned</span> <span class="type">int</span> <span class="title">DH::GetKey</span><span class="params">(<span class="type">const</span> <span class="type">unsigned</span> <span class="type">int</span> &amp;y, <span class="type">const</span> <span class="type">unsigned</span> <span class="type">int</span> &amp;x)</span> <span class="comment">// 参数：对方的公钥，本方的私钥</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="comment">// 密钥K = 对方的公钥Y的本方的私钥X次方 mod p</span></span><br><span class="line">    <span class="type">unsigned</span> <span class="type">int</span> k = <span class="keyword">this</span>-&gt;<span class="built_in">QuickPowMod</span>(y, x, <span class="keyword">this</span>-&gt;p_arg_);</span><br><span class="line"></span><br><span class="line">    <span class="keyword">return</span> k;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="comment">// 获取素数</span></span><br><span class="line"><span class="function"><span class="type">unsigned</span> <span class="type">int</span> <span class="title">DH::GetPrimeNum</span><span class="params">()</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="type">unsigned</span> <span class="type">int</span> random = <span class="number">0</span>;     <span class="comment">// 随机数</span></span><br><span class="line">    <span class="type">unsigned</span> <span class="type">int</span> random_odd = <span class="number">0</span>; <span class="comment">// 随机奇数</span></span><br><span class="line"></span><br><span class="line">    <span class="type">unsigned</span> <span class="type">int</span> n = <span class="number">0</span>;              <span class="comment">// 素性测试的参数n 循环中需要重新初始化</span></span><br><span class="line">    <span class="type">unsigned</span> <span class="type">int</span> a = <span class="number">0</span>;              <span class="comment">// 素性测试的参数a</span></span><br><span class="line">    <span class="type">bool</span> primality_test_res = <span class="literal">false</span>; <span class="comment">// 一次素性测试结果    false不是素数true可能为素数</span></span><br><span class="line">    <span class="type">bool</span> prime_flag = <span class="literal">false</span>;         <span class="comment">// 素数标志，最终素性测试结果。false0不是素数，true1可能为素数</span></span><br><span class="line">    <span class="comment">// 提示：初始化在循环外的变量在循环中注意是否需要更新、重新初始化</span></span><br><span class="line"></span><br><span class="line">    <span class="keyword">while</span> (<span class="number">1</span>) <span class="comment">// 循环</span></span><br><span class="line">    &#123;</span><br><span class="line">        <span class="comment">// 1.1随机取一个期望大小的奇数</span></span><br><span class="line">        <span class="comment">// 1.1.1取随机数</span></span><br><span class="line">        random = <span class="built_in">rand</span>(); <span class="comment">// 随机数 一般是4~5位数，不超过unsigned int的表示范围</span></span><br><span class="line"></span><br><span class="line">        <span class="comment">// 1.1.2取奇数</span></span><br><span class="line">        <span class="keyword">if</span> (random % <span class="number">2</span> == <span class="number">0</span>) <span class="comment">// 如果是偶数，+1成为奇数</span></span><br><span class="line">        &#123;</span><br><span class="line">            random_odd = random + <span class="number">1</span>;</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="keyword">else</span> <span class="comment">// 奇数不额外操作</span></span><br><span class="line">        &#123;</span><br><span class="line">            random_odd = random;</span><br><span class="line">        &#125;</span><br><span class="line"></span><br><span class="line">        <span class="comment">// 1.2使用素性测试判断</span></span><br><span class="line">        n = random_odd;</span><br><span class="line"></span><br><span class="line">        <span class="keyword">for</span> (<span class="type">int</span> i = <span class="number">0</span>; i &lt; <span class="number">128</span>; ++i) <span class="comment">// 选取128个参数a，测试128次</span></span><br><span class="line">        &#123;</span><br><span class="line">            <span class="comment">//  1.2.1随机选择相关参数a。满足a为整数，1 &lt; a &lt; n - 1</span></span><br><span class="line">            a = <span class="built_in">rand</span>() % (n - <span class="number">1</span>); <span class="comment">// 0 ~ n - 2</span></span><br><span class="line">            <span class="comment">// 注意：</span></span><br><span class="line">            <span class="comment">// 因为运行时间段相近，第一次a取的随机数可能和n相等</span></span><br><span class="line">            <span class="comment">// 则计算后结果必为1，而后1 + 1 = 2</span></span><br><span class="line">            <span class="comment">// 将设置随机种子代码提取出函数后，排除该错误</span></span><br><span class="line">            <span class="keyword">if</span> (a == <span class="number">0</span>) <span class="comment">// 如果是0，令a = 2 &gt; 1</span></span><br><span class="line">            &#123;</span><br><span class="line">                a += <span class="number">2</span>;</span><br><span class="line">            &#125;</span><br><span class="line">            <span class="keyword">if</span> (a == <span class="number">1</span>) <span class="comment">// 如果是1，令a = 2 &gt; 1</span></span><br><span class="line">            &#123;</span><br><span class="line">                ++a;</span><br><span class="line">            &#125;</span><br><span class="line"></span><br><span class="line">            primality_test_res = <span class="built_in">PrimalityTest</span>(random_odd, a); <span class="comment">// 素性测试</span></span><br><span class="line"></span><br><span class="line">            <span class="keyword">if</span> (primality_test_res == <span class="literal">true</span>) <span class="comment">// 一次测试结果可能为素数</span></span><br><span class="line">            &#123;</span><br><span class="line">                prime_flag = <span class="literal">true</span>; <span class="comment">// 标记可能为素数</span></span><br><span class="line">            &#125;</span><br><span class="line">            <span class="keyword">else</span> <span class="keyword">if</span> (primality_test_res == <span class="literal">false</span>) <span class="comment">// 只要有一次素性测试不是素数，则必不为素数</span></span><br><span class="line">            &#123;</span><br><span class="line">                prime_flag = <span class="literal">false</span>;</span><br><span class="line"></span><br><span class="line">                <span class="keyword">break</span>; <span class="comment">// 不再用a测试，需要重新选取随机奇数</span></span><br><span class="line">            &#125;</span><br><span class="line">        &#125;</span><br><span class="line"></span><br><span class="line">        <span class="keyword">if</span> (prime_flag == <span class="literal">true</span>) <span class="comment">// 随机奇数可能为素数，</span></span><br><span class="line">        &#123;</span><br><span class="line">            <span class="keyword">break</span>; <span class="comment">// 退出循环</span></span><br><span class="line">        &#125;</span><br><span class="line">        <span class="comment">// 否则随机奇数不是素数，进入循环，再重新进行1.1取随机奇数，1.2素性测试步骤</span></span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="keyword">return</span> random_odd; <span class="comment">// 获得素数</span></span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="comment">// Miller-Rabin素性测试</span></span><br><span class="line"><span class="function"><span class="type">bool</span> <span class="title">DH::PrimalityTest</span><span class="params">(<span class="type">const</span> <span class="type">unsigned</span> <span class="type">int</span> &amp;n, <span class="type">const</span> <span class="type">unsigned</span> <span class="type">int</span> &amp;a)</span> <span class="comment">// 参数：随机奇数，参数a</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="comment">// 1.2.2找到相关参数k，q。满足n - 1 = 2 ^ k × q。k、q为整数，k &gt; 0，q为奇数</span></span><br><span class="line">    <span class="type">unsigned</span> <span class="type">int</span> k = <span class="number">0</span>;</span><br><span class="line">    <span class="type">unsigned</span> <span class="type">int</span> q = n - <span class="number">1</span>;</span><br><span class="line"></span><br><span class="line">    <span class="comment">// 提示：</span></span><br><span class="line">    <span class="comment">// 很多算法都只说明要找到k、q，却不说怎么找</span></span><br><span class="line">    <span class="comment">// 找k，q的代码也含糊其辞的</span></span><br><span class="line">    <span class="keyword">while</span> ((q &amp; <span class="number">1</span>) == <span class="number">0</span>)</span><br><span class="line">    &#123;</span><br><span class="line">        ++k;</span><br><span class="line">        q &gt;&gt;= <span class="number">1</span>;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="comment">// 理解：</span></span><br><span class="line">    <span class="comment">// q &amp; 1：即q的二进制表示与二进制位1与运算，取q二进制表示的最低位0或1</span></span><br><span class="line">    <span class="comment">// 如101 &amp; 1 = 101 &amp; 001 = 001 = 1</span></span><br><span class="line">    <span class="comment">// 如0010 &amp; 1 = 0010 &amp; 0001 = 0</span></span><br><span class="line"></span><br><span class="line">    <span class="comment">// 在最低位中，基数2 ^ 0 = 1，所以如果值是0，则1 × 0 = 0为偶数；值是1，则1 × 1 = 1为奇数</span></span><br><span class="line">    <span class="comment">// 所以，如果运算结果为0，则是偶数，可以提取一个因子2</span></span><br><span class="line">    <span class="comment">// while：连续提取因子2</span></span><br><span class="line">    <span class="comment">// 每提取一个因子2，则++k，k是因子2的计数</span></span><br><span class="line">    <span class="comment">// q &gt;&gt;= 1：将q的二进制表示右移缩小，继续对最低位判断提取因子2</span></span><br><span class="line">    <span class="comment">// 直到不能连续提取因子2，则q即为所求</span></span><br><span class="line"></span><br><span class="line">    <span class="comment">// 如十进制13 - 1 = 12 = 二进制1100，在第1、2位提取因子2为2 ^ 2 = 4</span></span><br><span class="line">    <span class="comment">// 所以12 = 2 ^ 2 × 3。k = 2，q = 3</span></span><br><span class="line">    <span class="comment">// 如十进制7 - 1 = 6 = 二进制110，在第1位提取1个因子2为2 ^ 1 = 2</span></span><br><span class="line">    <span class="comment">// 所以6 = 2 ^ 1 × 3。k = 1，q = 3</span></span><br><span class="line"></span><br><span class="line">    <span class="comment">// 提示：注意k、q的取值条件</span></span><br><span class="line">    <span class="comment">// 对正整数素数，除了2为偶数，其他数必为奇数</span></span><br><span class="line">    <span class="comment">// 奇数-1必为偶数，必至少能提取1个公因子2，则k至少为1 &gt; 0满足</span></span><br><span class="line">    <span class="comment">// 由算法性质，知提取所有的公因子2，则结果q必为奇数满足</span></span><br><span class="line">    <span class="comment">// 一般q数很大，所以在接下来的步骤需要用蒙哥马利快速模幂算法</span></span><br><span class="line"></span><br><span class="line">    <span class="comment">// 1.2.3计算a ^ q % n</span></span><br><span class="line">    <span class="type">unsigned</span> <span class="type">int</span> aq_mod_n = <span class="keyword">this</span>-&gt;<span class="built_in">QuickPowMod</span>(a, q, n);</span><br><span class="line"></span><br><span class="line">    <span class="comment">// cout &lt;&lt; n &lt;&lt; endl;</span></span><br><span class="line">    <span class="comment">// cout &lt;&lt; k &lt;&lt; endl;</span></span><br><span class="line">    <span class="comment">// cout &lt;&lt; q &lt;&lt; endl;</span></span><br><span class="line">    <span class="comment">// cout &lt;&lt; a &lt;&lt; endl;</span></span><br><span class="line">    <span class="comment">// cout &lt;&lt; aq_mod_n &lt;&lt; endl;</span></span><br><span class="line"></span><br><span class="line">    <span class="comment">// 1.2.4运用二次探测定理的逆否命题判断</span></span><br><span class="line">    <span class="comment">// 正命题大概：探测，所有解只有1或n-1，则可能为素数</span></span><br><span class="line">    <span class="comment">// 逆否命题大概：探测，存在解不为1且不为n-1，则必定不是素数</span></span><br><span class="line">    <span class="comment">// 可以用正命题也可以用逆否命题判断。以下用正命题和逆否命题判断</span></span><br><span class="line">    <span class="comment">// 第一个判断条件：未探测时，a ^ q % n == 1，则可能为素数</span></span><br><span class="line">    <span class="keyword">if</span> (aq_mod_n == <span class="number">1</span>)</span><br><span class="line">    &#123;</span><br><span class="line">        <span class="keyword">return</span> <span class="literal">true</span>;</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="comment">// 第二个判断条件：二次探测时，只要存在不为1且不为n-1，则必定不是素数</span></span><br><span class="line">    <span class="keyword">for</span> (<span class="type">int</span> j = <span class="number">0</span>; j &lt; k; ++j) <span class="comment">// 0 ~ k-1</span></span><br><span class="line">    &#123;</span><br><span class="line">        aq_mod_n = <span class="keyword">this</span>-&gt;<span class="built_in">QuickPowMod</span>(aq_mod_n, <span class="number">2</span>, n);</span><br><span class="line">        <span class="comment">// 对序列二次探测 计算a ^ (q × 2 ^ j) % n = aq_mod_n ^ (2 ^ j) % n。每次循环都幂2相当于(2 ^ j)</span></span><br><span class="line"></span><br><span class="line">        <span class="keyword">if</span> (aq_mod_n != <span class="number">1</span> &amp;&amp; aq_mod_n != n - <span class="number">1</span>)</span><br><span class="line">        &#123;</span><br><span class="line">            <span class="keyword">return</span> <span class="literal">false</span>;</span><br><span class="line">        &#125;</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="keyword">return</span> <span class="literal">true</span>; <span class="comment">// 第二个判断条件：二次探测时，若没有因判断为合数而返回，则可能为素数</span></span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="comment">// 蒙哥马利快速幂模运算</span></span><br><span class="line"><span class="comment">// 参数：a ^ q % n</span></span><br><span class="line"><span class="comment">// 返回值：a ^ q % n</span></span><br><span class="line"><span class="function"><span class="type">unsigned</span> <span class="type">int</span> <span class="title">DH::QuickPowMod</span><span class="params">(<span class="type">const</span> <span class="type">unsigned</span> <span class="type">int</span> &amp;a, <span class="type">const</span> <span class="type">unsigned</span> <span class="type">int</span> &amp;q, <span class="type">const</span> <span class="type">unsigned</span> <span class="type">int</span> &amp;n)</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="comment">// 原理：</span></span><br><span class="line">    <span class="comment">//  幂运算性质：a ^ q = a ^ q1 × a ^ q2。q = q1 + q2</span></span><br><span class="line">    <span class="comment">//  模运算性质：(a × b) % n = [(a % n) × (b % n)] % n</span></span><br><span class="line">    <span class="comment">// 所以：a ^ q % n = (a ^ q1 × a ^ q2) % n = [(a ^ q1 % n) × (a ^ q2 % n)] % n</span></span><br><span class="line">    <span class="type">unsigned</span> <span class="type">int</span> res = <span class="number">1</span>;</span><br><span class="line">    <span class="type">unsigned</span> <span class="type">int</span> a_temp = a; <span class="comment">// 运算中会改变a的值，暂存</span></span><br><span class="line">    <span class="type">unsigned</span> <span class="type">int</span> q_temp = q; <span class="comment">// 运算中会改变q的值，暂存</span></span><br><span class="line"></span><br><span class="line">    <span class="comment">// 提示：很多算法代码含糊其辞的</span></span><br><span class="line">    <span class="keyword">while</span> (q_temp &gt; <span class="number">0</span>)</span><br><span class="line">    &#123;</span><br><span class="line">        <span class="keyword">if</span> ((q_temp &amp; <span class="number">1</span>) == <span class="number">1</span>)</span><br><span class="line">        &#123;</span><br><span class="line">            res = <span class="keyword">this</span>-&gt;<span class="built_in">QuickMulMod</span>(res, a_temp, n);</span><br><span class="line">        &#125;</span><br><span class="line"></span><br><span class="line">        a_temp = <span class="keyword">this</span>-&gt;<span class="built_in">QuickMulMod</span>(a_temp, a_temp, n);</span><br><span class="line"></span><br><span class="line">        q_temp &gt;&gt;= <span class="number">1</span>;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="comment">// 理解：</span></span><br><span class="line">    <span class="comment">// 算法是针对十进制数的二进制表示进行运算的</span></span><br><span class="line"></span><br><span class="line">    <span class="comment">// while (q_temp &gt; 0)：对比素性测试的内容：while ((q &amp; 1) == 0)</span></span><br><span class="line">    <span class="comment">// 这里是判断值，需要判断所有二进制位，所以只要q在后面的右移位中值不为0，就循环。而素性测试中是判断位</span></span><br><span class="line"></span><br><span class="line">    <span class="comment">// if ((q_temp &amp; 1) == 1)：最低位为1时，该位有效，需要计算并更新结果</span></span><br><span class="line">    <span class="comment">// 快速乘算法：res = (res × a_temp) % n</span></span><br><span class="line">    <span class="comment">// 该步骤相当于每次计算单个的(a ^ q2 % n)，然后和之前的(a ^ q1 % n)相乘作为新的结果</span></span><br><span class="line">    <span class="comment">// 其中第一个res是更新结果，第二个res是之前的结果，a_temp是当前的基数</span></span><br><span class="line">    <span class="comment">// 基数：在循环中对每一位都会更新基数（见后面步骤），在二进制表示为1时，该基数有效</span></span><br><span class="line"></span><br><span class="line">    <span class="comment">// a_temp = QuickMulMod(a_temp, a_temp, n);相当于a_temp = a_temp × a_temp % n</span></span><br><span class="line">    <span class="comment">// 如初始a_temp = 2，则不断更新为2 ^ 0 = 1，2 ^ 1 = 2</span></span><br><span class="line">    <span class="comment">// 再进行%保证基数不超过范围</span></span><br><span class="line"></span><br><span class="line">    <span class="keyword">return</span> res;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="comment">// 快速乘</span></span><br><span class="line"><span class="comment">// 参数：a * b % c</span></span><br><span class="line"><span class="comment">// 返回值：a * b % c</span></span><br><span class="line"><span class="function"><span class="type">unsigned</span> <span class="type">int</span> <span class="title">DH::QuickMulMod</span><span class="params">(<span class="type">const</span> <span class="type">unsigned</span> <span class="type">int</span> &amp;a, <span class="type">const</span> <span class="type">unsigned</span> <span class="type">int</span> &amp;b, <span class="type">const</span> <span class="type">unsigned</span> <span class="type">int</span> &amp;c)</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="comment">// 原理：</span></span><br><span class="line">    <span class="comment">// 同快速幂模运算，将乘法转换为加法运算</span></span><br><span class="line">    <span class="comment">// a × b % c = [(a + a) % c] + [(a + a) % c] + ... [(a + a) % c]共b个a相加求模</span></span><br><span class="line">    <span class="type">unsigned</span> <span class="type">int</span> res = <span class="number">0</span>;</span><br><span class="line">    <span class="type">unsigned</span> <span class="type">int</span> a_temp = a;</span><br><span class="line">    <span class="type">unsigned</span> <span class="type">int</span> b_temp = b;</span><br><span class="line"></span><br><span class="line">    <span class="keyword">while</span> (b_temp &gt; <span class="number">0</span>)</span><br><span class="line">    &#123;</span><br><span class="line">        <span class="keyword">if</span> (b_temp &amp; <span class="number">1</span>)</span><br><span class="line">        &#123;</span><br><span class="line">            res = (res + a_temp) % c;</span><br><span class="line">        &#125;</span><br><span class="line"></span><br><span class="line">        a_temp = (a_temp + a_temp) % c;</span><br><span class="line"></span><br><span class="line">        b_temp &gt;&gt;= <span class="number">1</span>;</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="keyword">return</span> res;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="comment">// 获取本原根</span></span><br><span class="line"><span class="function"><span class="type">unsigned</span> <span class="type">int</span> <span class="title">DH::GetPrimitiveRoot</span><span class="params">()</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="comment">// 大致思路：对素数p，p的欧拉函数p - 1，底数a属于（1，p），指数m属于（1，p）</span></span><br><span class="line">    <span class="comment">// 正向判定本原根满足：</span></span><br><span class="line">    <span class="comment">// 1.当m = （1，p - 1）的数时，所有a^m % p 不等于 1</span></span><br><span class="line">    <span class="comment">// 2.当m = p - 1时，a^m % p = 1</span></span><br><span class="line">    <span class="comment">// 则a为本原根</span></span><br><span class="line"></span><br><span class="line">    <span class="comment">// 逆向否定本原根满足：由正向的1.逆否命题得</span></span><br><span class="line">    <span class="comment">// 1.当m = （1，p - 1）的数时，存在a^m % p = 1，则a不为本原根</span></span><br><span class="line"></span><br><span class="line">    <span class="comment">// 快速正向判定本原根满足：</span></span><br><span class="line">    <span class="comment">// 1.当m = （p - 1的素因子）的数时，所有a^m % p 不等于 1，则a为本原根</span></span><br><span class="line"></span><br><span class="line">    <span class="comment">// 慢速正向判定本原根满足：由快速正向扩大搜索范围得</span></span><br><span class="line">    <span class="comment">// 1.当m = （1，p - 1）的数时，所有a^m % p 不等于 1，则a为本原根</span></span><br><span class="line"></span><br><span class="line">    <span class="comment">// 提示：</span></span><br><span class="line">    <span class="comment">// 可以证明暴力枚举求最小本原根的复杂度是可以接受的，采用正向逆向结合判定本原根的暴力枚举法</span></span><br><span class="line">    <span class="comment">// 参数p的欧拉函数为p -1</span></span><br><span class="line">    <span class="type">unsigned</span> <span class="type">int</span> calcul_res = <span class="number">0</span>; <span class="comment">// 计算结果</span></span><br><span class="line"></span><br><span class="line">    <span class="comment">// 2.1遍历（1，p）为底数</span></span><br><span class="line">    <span class="keyword">for</span> (<span class="type">unsigned</span> <span class="type">int</span> i = <span class="number">2</span>; i &lt; <span class="keyword">this</span>-&gt;p_arg_; ++i) <span class="comment">// 范围：2 ~ p - 1</span></span><br><span class="line">    &#123;</span><br><span class="line">        <span class="comment">// 2.2遍历（1，p）为指数</span></span><br><span class="line">        <span class="keyword">for</span> (<span class="type">unsigned</span> <span class="type">int</span> j = <span class="number">2</span>; j &lt; <span class="keyword">this</span>-&gt;p_arg_; ++j) <span class="comment">// 范围：2 ~ p - 1</span></span><br><span class="line">        &#123;</span><br><span class="line">            calcul_res = <span class="keyword">this</span>-&gt;<span class="built_in">QuickPowMod</span>(i, j, <span class="keyword">this</span>-&gt;p_arg_); <span class="comment">// 计算结果 = i ^ j % p</span></span><br><span class="line"></span><br><span class="line">            <span class="comment">// 2.2.1当m = （1，p - 1）的数时，所有a^m % p 不等于 1</span></span><br><span class="line">            <span class="comment">// 逆否命题为当m = （1，p - 1）的数时，存在a^m % p = 1，则a不为本原根</span></span><br><span class="line">            <span class="keyword">if</span> ((j != <span class="keyword">this</span>-&gt;p_arg_ - <span class="number">1</span>) &amp;&amp; (calcul_res == <span class="number">1</span>))</span><br><span class="line">            &#123;</span><br><span class="line">                <span class="keyword">break</span>; <span class="comment">// 退出j循环，取下一个i</span></span><br><span class="line">            &#125;</span><br><span class="line"></span><br><span class="line">            <span class="comment">// 2.2.2在2.2.1没退出j循环，且当m = p - 1时，a ^ m % p = 1</span></span><br><span class="line">            <span class="keyword">if</span> ((j == <span class="keyword">this</span>-&gt;p_arg_ - <span class="number">1</span>) &amp;&amp; (calcul_res == <span class="number">1</span>))</span><br><span class="line">            &#123;</span><br><span class="line">                <span class="keyword">return</span> i; <span class="comment">// i为（最小）本原根，返回本原根i</span></span><br><span class="line">            &#125;</span><br><span class="line">        &#125;</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="comment">// 提示：素数必存在本原根，存在i并返回。但代码逻辑需考虑不满足if()条件返回i的另外返回条件</span></span><br><span class="line">    <span class="keyword">return</span> <span class="number">0</span>; <span class="comment">// 必不会执行</span></span><br><span class="line">&#125;</span><br><span class="line"></span><br></pre></td></tr></table></figure>
<hr>
<h2 id="main-cpp"><a href="#main-cpp" class="headerlink" title="main.cpp"></a>main.cpp</h2><figure class="highlight cpp"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">#<span class="keyword">include</span> <span class="string">&quot;dh.h&quot;</span></span></span><br><span class="line"></span><br><span class="line"><span class="function"><span class="type">int</span> <span class="title">main</span><span class="params">(<span class="type">int</span> argc, <span class="type">char</span> **argv)</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    DH *dh = <span class="keyword">new</span> <span class="built_in">DH</span>(); <span class="comment">// DH对象</span></span><br><span class="line"></span><br><span class="line">    <span class="comment">// 用户A</span></span><br><span class="line">    <span class="type">unsigned</span> <span class="type">int</span> xa = dh-&gt;<span class="built_in">GetPrivateKey</span>(); <span class="comment">// 1.获取私钥</span></span><br><span class="line">    cout &lt;&lt; <span class="string">&quot;用户A的密钥生成：\t&quot;</span> &lt;&lt; endl;</span><br><span class="line">    cout &lt;&lt; <span class="string">&quot;私钥xa：\t&quot;</span> &lt;&lt; xa &lt;&lt; endl;</span><br><span class="line"></span><br><span class="line">    <span class="type">unsigned</span> <span class="type">int</span> ya = dh-&gt;<span class="built_in">GetPublicKey</span>(xa); <span class="comment">// 2.获取公钥</span></span><br><span class="line">    cout &lt;&lt; <span class="string">&quot;公钥ya：\t&quot;</span> &lt;&lt; ya &lt;&lt; endl;</span><br><span class="line">    cout &lt;&lt; endl;</span><br><span class="line"></span><br><span class="line">    <span class="comment">// 用户B</span></span><br><span class="line">    <span class="type">unsigned</span> <span class="type">int</span> xb = dh-&gt;<span class="built_in">GetPrivateKey</span>(); <span class="comment">// 1.获取私钥</span></span><br><span class="line">    cout &lt;&lt; <span class="string">&quot;用户B的密钥生成：\t&quot;</span> &lt;&lt; endl;</span><br><span class="line">    cout &lt;&lt; <span class="string">&quot;私钥xb：\t&quot;</span> &lt;&lt; xb &lt;&lt; endl;</span><br><span class="line"></span><br><span class="line">    <span class="type">unsigned</span> <span class="type">int</span> yb = dh-&gt;<span class="built_in">GetPublicKey</span>(xb); <span class="comment">// 2.获取公钥</span></span><br><span class="line">    cout &lt;&lt; <span class="string">&quot;公钥yb：\t&quot;</span> &lt;&lt; yb &lt;&lt; endl;</span><br><span class="line">    cout &lt;&lt; endl;</span><br><span class="line"></span><br><span class="line">    <span class="comment">// 用户A</span></span><br><span class="line">    <span class="type">unsigned</span> <span class="type">int</span> ka = dh-&gt;<span class="built_in">GetKey</span>(yb, xa); <span class="comment">// 3.密钥生成</span></span><br><span class="line">    cout &lt;&lt; <span class="string">&quot;用户A计算产生密钥：\t&quot;</span> &lt;&lt; endl;</span><br><span class="line">    cout &lt;&lt; <span class="string">&quot;密钥ka：\t&quot;</span> &lt;&lt; ka &lt;&lt; endl;</span><br><span class="line">    cout &lt;&lt; endl;</span><br><span class="line"></span><br><span class="line">    <span class="comment">// 用户B</span></span><br><span class="line">    <span class="type">unsigned</span> <span class="type">int</span> kb = dh-&gt;<span class="built_in">GetKey</span>(ya, xb); <span class="comment">// 3.密钥生成</span></span><br><span class="line">    cout &lt;&lt; <span class="string">&quot;用户B计算产生密钥：\t&quot;</span> &lt;&lt; endl;</span><br><span class="line">    cout &lt;&lt; <span class="string">&quot;密钥kb：\t&quot;</span> &lt;&lt; kb &lt;&lt; endl;</span><br><span class="line"></span><br><span class="line">    <span class="keyword">delete</span> dh;</span><br><span class="line"></span><br><span class="line">    <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>
<hr>
<h1 id="结果"><a href="#结果" class="headerlink" title="结果"></a>结果</h1><p><img src= "" data-lazy-src="/../img/dh/5c8347e61c6242dca83763e0c30e95c8.png" alt="在这里插入图片描述"></p>
<hr>
<h1 id="总结"><a href="#总结" class="headerlink" title="总结"></a>总结</h1><p>网络安全中Diffie-Hellman的C++语言描述简单实现。</p>
<hr>
<h1 id="参考资料"><a href="#参考资料" class="headerlink" title="参考资料"></a>参考资料</h1><ul>
<li>《密码编码学与网络安全——原理与实践(第五版)》作者：William Stallings</li>
<li><a target="_blank" rel="noopener" href="https://www.cnblogs.com/pam-sh/p/16491643.html">求本原根 - PamShao - 博客园 (cnblogs.com)</a></li>
<li><a target="_blank" rel="noopener" href="https://www.cnblogs.com/Dance-Of-Faith/p/9905786.html">【科技】原根的快速判断方法以及对1求离散对数的另一种想法 - Dance_Of_Faith - 博客园 (cnblogs.com)</a></li>
<li><a target="_blank" rel="noopener" href="https://oi-wiki.org/math/number-theory/primitive-root/">原根 - OI Wiki (oi-wiki.org</a></li>
</ul>
<hr>
<h1 id="作者的话"><a href="#作者的话" class="headerlink" title="作者的话"></a>作者的话</h1><ul>
<li><strong>感谢参考资料的作者&#x2F;博主</strong></li>
<li>作者：夜悊</li>
<li>版权所有，转载请注明出处，谢谢~</li>
<li><strong>如果文章对你有帮助，请点个赞或加个粉丝吧，你的支持就是作者的动力~</strong></li>
<li>文章在描述时有疑惑的地方，请留言，定会一一耐心讨论、解答</li>
<li>文章在认识上有错误的地方, 敬请批评指正</li>
<li>望读者们都能有所收获</li>
</ul>
<hr>
</article><div class="post-copyright"><div class="post-copyright__author"><span class="post-copyright-meta"><i class="fas fa-circle-user fa-fw"></i>文章作者: </span><span class="post-copyright-info"><a href="http://example.com">夜悊</a></span></div><div class="post-copyright__type"><span class="post-copyright-meta"><i class="fas fa-square-arrow-up-right fa-fw"></i>文章链接: </span><span class="post-copyright-info"><a href="http://example.com/2022/11/16/Diffie-Hellman%E7%9A%84C++%E8%AF%AD%E8%A8%80%E6%8F%8F%E8%BF%B0%E7%AE%80%E5%8D%95%E5%AE%9E%E7%8E%B0/">http://example.com/2022/11/16/Diffie-Hellman的C++语言描述简单实现/</a></span></div><div class="post-copyright__notice"><span class="post-copyright-meta"><i class="fas fa-circle-exclamation fa-fw"></i>版权声明: </span><span class="post-copyright-info">本博客所有文章除特别声明外，均采用 <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/" target="_blank">CC BY-NC-SA 4.0</a> 许可协议。转载请注明来自 <a href="http://example.com" target="_blank">夜悊的技术小宅</a>！</span></div></div><div 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href="/2022/10/24/%E6%95%B0%E6%8D%AE%E5%8A%A0%E5%AF%86%E6%A0%87%E5%87%86%EF%BC%88DES%EF%BC%89%E7%9A%84C++%E8%AF%AD%E8%A8%80%E6%8F%8F%E8%BF%B0%E5%AE%9E%E7%8E%B0/" title="数据加密标准（DES）的C++语言描述实现"><img class="cover" src= "" data-lazy-src="/img/cover10.png" alt="cover"><div class="content is-center"><div class="date"><i class="far fa-calendar-alt fa-fw"></i> 2022-10-24</div><div class="title">数据加密标准（DES）的C++语言描述实现</div></div></a></div></div></div></div><div class="aside-content" id="aside-content"><div class="sticky_layout"><div class="card-widget" id="card-toc"><div class="item-headline"><i class="fas fa-stream"></i><span>目录</span><span class="toc-percentage"></span></div><div class="toc-content is-expand"><ol class="toc"><li class="toc-item toc-level-1"><a class="toc-link" href="#%E5%89%8D%E8%A8%80"><span class="toc-number">1.</span> <span class="toc-text">前言</span></a></li><li class="toc-item toc-level-1"><a class="toc-link" href="#%E4%BB%A3%E7%A0%81%E4%BB%93%E5%BA%93"><span class="toc-number">2.</span> <span class="toc-text">代码仓库</span></a></li><li class="toc-item toc-level-1"><a class="toc-link" href="#%E4%BB%A3%E7%A0%81%E7%89%B9%E7%82%B9"><span class="toc-number">3.</span> <span class="toc-text">代码特点</span></a></li><li class="toc-item toc-level-1"><a class="toc-link" href="#%E4%BB%A3%E7%A0%81"><span class="toc-number">4.</span> <span class="toc-text">代码</span></a><ol class="toc-child"><li class="toc-item toc-level-2"><a class="toc-link" href="#dh-h"><span class="toc-number">4.1.</span> <span class="toc-text">dh.h</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#dh-cpp"><span class="toc-number">4.2.</span> <span class="toc-text">dh.cpp</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#main-cpp"><span class="toc-number">4.3.</span> <span class="toc-text">main.cpp</span></a></li></ol></li><li class="toc-item toc-level-1"><a class="toc-link" href="#%E7%BB%93%E6%9E%9C"><span class="toc-number">5.</span> <span class="toc-text">结果</span></a></li><li class="toc-item toc-level-1"><a class="toc-link" href="#%E6%80%BB%E7%BB%93"><span class="toc-number">6.</span> <span class="toc-text">总结</span></a></li><li class="toc-item toc-level-1"><a class="toc-link" href="#%E5%8F%82%E8%80%83%E8%B5%84%E6%96%99"><span class="toc-number">7.</span> <span class="toc-text">参考资料</span></a></li><li class="toc-item toc-level-1"><a class="toc-link" href="#%E4%BD%9C%E8%80%85%E7%9A%84%E8%AF%9D"><span class="toc-number">8.</span> <span class="toc-text">作者的话</span></a></li></ol></div></div></div></div></main><footer id="footer" style="background-image: url('/img/cover6.png')"><div id="footer-wrap"><div class="copyright">&copy;2023 - 2025 By 夜悊</div><div class="footer_custom_text"><a target="_blank" href="https://beian.miit.gov.cn/" >琼ICP备2023001225号-1</a><br><img src= "" data-lazy-src="/img/beian.png"/>&nbsp<a target="_blank" href="https://www.beian.gov.cn/portal/registerSystemInfo?recordcode=46010802001221" >琼公网安备 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